patrizio angelini, giordano da lozzo, giuseppe di battista...
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Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista,
Fabrizio Frati, Maurizio Patrignani, Vincenzo Roselli
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Graph G = (V,E)
Vertices are points
Edges are straight-line segments
Planar: edges do not cross each other
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Metamorphosis I -- M. C. Escher
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Morphing is a special effect in motion pictures and animations that changes (or morphs) one image or shape into another through a seamless transition.
(Wikipedia)
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… changes (or morphs) one planar graph drawing into another through a seamless transition
Vertices move at uniform speed along straight-line segments
Planarity is preserved during the whole morph
(…)
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Some pairs of drawings require several steps
The complexity of a morphing algorithm is defined as the number of steps it performs
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Tietze, Rend. Circ. Matematico di Palermo, 38(1):247-304, 1914
We are in the centennial of the Morphing problem!
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Smith, 1917
Veblen, 1917
Alexander, 1923
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Cairns, 1944
A planar morph exists between any two planar drawings of any maximal planar graph(with an exponential number of steps)
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Thomassen, Journal of Combinatorial Theory B, 1983
A planar convex morph exists between any two planar convex drawings of any maximal planar graph,
(with an exponential number of steps)
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Aicholzer et al, 2011
Floater, Gotsman, Surazhskyseveral papers
Lubiw Petrick, 2011
Biedl et al., 2006
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Alamdari et al., SODA 2013
A planar morph exists between any two planar drawings of any maximal planar graph,
(with a polynomial numer of steps)
O(n2) steps for maximal planar
O(n4) steps for maximal planar
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Angelini et al., GD 2013
Barrera-Cruz et al., MCDMCG, 2013
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Angelini et al., GD 2013
Barrera-Cruz et al., MCDMCG, 2013
At each morphing step, all the vertices move along the same direction
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Angelini et al., GD 2013
Barrera-Cruz et al., MCDMCG, 2013
At each morphing step, all the vertices move along the same direction
Maximal Planar graphs in O(n2) stepsSeries-parallel graphs in O(n) steps
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Theorem 1: There exist an algorithm to morph planar graph drawings in O(n) steps
Theorem 2: There exist two drawings of a planar graph (a path) such that morphing one into the other requires at least Ω(n) steps
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Every planar graph contains at least a vertex of degree at most 5 with no chord between its neighbors
By Euler’s formula, |E| <= 3n-6
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Every polygon with at most 5 vertices has at least one vertex on the boundary of its kernel
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Every polygon with at most 5 vertices has at least one vertex on the boundary of its kernel
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Contraction of a low-degree vertex on one of its neighbors
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1: Select a vertex v of low degree.
It might be contractibleon different neighborsin the two drawings
vv
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2: Contract v in boththe drawings.
We obtain differentgraphs!
vv
O(1) O(1)
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3: Remove v and compute a drawing in which the polygon of itsneighbors is convex
vv
O(1) O(1)
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In this drawing, passing from one neighbor to the other is always possible!
vv
O(1) O(1)
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In this drawing, passing from one neighbor to the other is always possible!
vv
O(1) O(1)
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In this drawing, passing from one neighbor to the other is always possible!
vv
O(1) O(1)
O(1)
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Recursively morph the two contracted drawings to the convex ones in T(n-1) steps
T(n) = 2 T(n-1) + O(1)
= O(2n)
vv
T(n-1) T(n-1)
O(1) O(1)
O(1)
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The morph is not to a specific convex drawing!v
v
O(n) T(n-1)
O(1) O(1)
O(1)
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The morph is not to a specific convex drawing!
T(n) = T(n-1) + O(n)
= O(n2)
vv
T(n-1)
O(1) O(1)
O(n)O(1)
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The morph is to a (very) specific convex drawing!v
v
O(1) T(n-1)
O(1) O(1)
O(1)
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The morph is to a (very) specific convex drawing!
T(n) = T(n-1) + O(1)
= O(n)
vv
T(n-1)
O(1) O(1)
O(1)O(1)
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Given:
a triconnected planar graph G = (V,E)
A set L of parallel lines
A mapping of the vertices of V to lines of L such that orienting the edges of E according to the order of the lines in L yields an st-orientation of G
st-orientation: each face is composed of two oriented paths
Then, G admits a convex drawing in which each vertex of V lies on the line of L it is mapped to
Hong, Nagamochi. J. of Discrete Algorithms, 2010
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Hong, Nagamochi. J. of Discrete Algorithms, 2010
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A linear number of steps are necessary to morph one of these drawings into the other
There exists an edge that must perform a linear number of complete rotations
In a single step, only (a bit less than) half rotation can be performed
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All the algorithms produce intermediate drawings on a very large grid
What about morphing on a limited-size grid?
Can we avoid contractions?
Can we produce convexity-preserving morphings with a polynomial number of steps?
How to morph non-planar graphs?
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